Optimal. Leaf size=2 \[ \tanh ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {28, 21, 207} \begin {gather*} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 28
Rule 207
Rubi steps
\begin {align*} \int \frac {1-x^2}{1-2 x^2+x^4} \, dx &=\int \frac {1-x^2}{\left (-1+x^2\right )^2} \, dx\\ &=-\int \frac {1}{-1+x^2} \, dx\\ &=\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] time = 0.00, size = 19, normalized size = 9.50 \begin {gather*} \frac {1}{2} \log (x+1)-\frac {1}{2} \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^2}{1-2 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.07, size = 13, normalized size = 6.50 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 15, normalized size = 7.50 \begin {gather*} \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 3, normalized size = 1.50 \begin {gather*} \arctanh \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.07, size = 13, normalized size = 6.50 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.30, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {atanh}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 12, normalized size = 6.00 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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